208 UNDULATORY THEORY OF LIGHT. 



And by pursuing a similar course with the values of the components CD, 

 OF, of the extraordinary ray, we shall obtain for its resultant intensity the two 

 values, also equivalent to each other — 



A'2 = V2rsinV+[sin2(2a-/5)-sin2/S]sin2-'^1 [39.] 



A'^^V^Tsin^;? -[cos2(2a-/5)-cos2/?]siu2-n [40.-] 



The intensity of the light in either plane is thus expressed in a formula of 

 two terms, one of which is affected by interference, while the other is not. It 

 is from the second that the colorific effects proceed — directly, when this term is 

 positive relatively to the first, and indirectly when it is relatively negative. 

 This term may therefore be called the chromatic term, and the other the achro- 

 matic. 



In considering these equations we observe, first, that if we add either valite 

 of A^ to either value of A'^, the chromatic tei-m disappears. The colors arc 

 therefore complementary; and if blended, the resultant is white. 



Secondly, since A^+A'^^zV^l^cos^/J-f-sinV)— V^ the sum of the two intensi- 

 ties is equal to the intensity of the original ray ; as it ought to be on the princi- 

 ple of the preservation of living forces. 



k 

 Thirdly, the chromatic effects being dependent on the factor sin^- for their 



character, will bo dependfiut not only upon this factor, but also on the coefficient, 

 sin^t 2(/. — \i) — sin^/3, or cos^(2a — Q) — cosV, for their quantity. Their greatest values 

 will hence occur when this coefBcient is maximum. There being two variables, 

 a and /?, if we make the first constant, wc shall find maxima when cos2(a — /?)=:0, 

 or cos2(/? — a)=0. This gives a series of values for 2(a — ?) or 2(/3 — a.\, 

 which are 90° and its odd multiples. It is sufficient to consider the first, which 

 gives a — ,'3=45°, or '} — a=45°; from which c=i|5+45°, or a=i{i — 45°. For 

 the higher values we need only replace 45° by the numbers 135°, 225'^, 315°, 

 &c., successively. These values substituted in the coefficient all give the same 

 result; hence all the maxima dependent on /3 alone are equal; and it is obvious 

 that they are independent of a, since a is not a function of /3. If we find, then, 

 the maximum with reference to a, and substitute in the resulting expression, 

 instead of /3, its value =:art45°, as obtained above, we shall have the maximum 

 of the maxima, or the azimuths of Q.Q' and 00', in which the chromatic effects 

 are the most brilliant possible. The solution gives sin(4a — 2/3)n=0. Hence 

 4a — 2/?=0, or 1S0°, or 360°, &c. Contenting ourselves with the first value, 

 and substituting for /3, we have 4« — 2o(^90°^0, or a=rrT45°. And as /j^ai 45°. 

 we conclude that the arrangement in which the colors will be most brilliant 

 is that in which the principal plane of the lamina is inclined 45° to the plane of 

 polarization of the incident light, and in which the principal plane of the 

 analyzer is in azimuth 0° or 90° — theoretic conclusions already anticipated 

 by experiment. 



Fourth, attending to the first of the formula3 for A^ and A'^ we see that the 

 chromatic term in each is symbolically positive. If the term, therefore, is essen- 

 tially positive in itself, the color of the ray is the color which the interference 

 expressed by that term would produce, diluted with such an amount of white 

 light as is expressed by the achromatic term. When the chromatic term in th(! 

 same formulaj becomes essentially negative, the color will be that which is left by 

 subtracting its own color from the amount of white in the achromatic term. 

 That the subtraction will be possible — that is to say, that, when the chromatic 

 term is negative, the achromatic term will always be the greater — will be evident 

 on inspection. For examining the coefficient of the chromatic term within the 

 bracket, it Avill be seen to consist of apositive and negative element, which elements. 



